Collins
left06800850137414000-1428751136650Collins A-level Mathematics Year 1 and ASScheme of Work 00Collins A-level Mathematics Year 1 and ASScheme of Work Scheme of WorkCollins A-level Mathematics Year 1 and ASThis course covers the requirements of the first year of the 2017 AS and A-level Mathematics specification. This scheme of work is designed to accompany the use of Collins’ A-level Mathematics Year 1 and AS Student Book, and references to sections in that book are given for each lesson.The scheme of work follows the order of the Student Book: this is not a suggested teaching order but is designed for flexible planning. Each chapter from the Student Book is divided into one-hour lessons. We have assumed that approximately 120 one-hour lessons are taught during the year, however lessons can be combined as you choose depending on your timetabling and preferred teaching order, and we have left some free time for additional linking, problem-solving or investigation work and assessments. We have suggested at the start of each chapter which chapters it would be beneficial to have already covered, to get the most out of the materials. In addition, we have brought out links and cross-references between different topics in column 4 to assist you with planning your own route through the course and making synoptic connections between topics for students.Chapters 1 – 11 cover the pure mathematics content that is tested at AS and also at A-level.Chapters 12 – 14 cover the statistics content that is tested at AS and also at A-level.Chapters 15 – 16 cover the mechanics content that is tested at AS and also at A-level.Each lesson is matched to the DfE-specified A-level Mathematics subject content and the corresponding Student Book section with all relevant exercises and activities listed. Learning objectives for each lesson are given, and a brief description of the lesson’s coverage to aid with your planning. A review lesson with assessment opportunities is plotted at the end of each chapter, with exam-style practice available from the Student Book and our Collins Connect digital platform.At the start of each chapter, we have suggested opportunities for you to use technology in the classroom. These ideas are developed further in the Student Book with in-context Technology boxes. In addition, you’ll find activities in the Student Book to explore the exam board-provided large data sets using a range of technologies. These are clearly indicated in the scheme of work, and can be flexibly adapted to be the focus of class, group or individual work.The scheme suggested is of course flexible and editable to correspond with your timetabling and to enable you to plan your own route through the course. We hope that you will find this a useful aid to your planning and teaching.KEYEx – exerciseDS – Data set activity (Statistics chapters only)Scheme of WorkCollins A-level Mathematics Year 1 and AS (120 hours)CHAPTER 1 – Algebra and functions 1: Manipulating algebraic expressions (8 Hours) Prior knowledge neededGCSE Maths onlyTechnologyUse the factorial and combinations button on a calculator.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 Manipulating polynomials algebraically Students collect like terms with expressions that include fractional coefficients and are expected to write expressions with terms written in descending order of index. They expand a single term over a bracket and expand two binomials. They use the same techniques to go on to expand three simple binomials.Manipulate polynomials algebraically including:expanding brackets and simplifying by collecting like termsA1 – mathematical proofB6 – manipulate polynomials algebraicallyLinks back to GCSE1.1Ex 1.1A2 Expanding multiple binomialsStudents are introduced to Pascal’s triangle and use this to expand multiple binomials. One check students can use when expanding multiple binomials is to ensure the sum of the indices in each term is the total of the index of the original expansion. Students use factorial notation when solve problems involving combinations.Understand and use the notation n! and nCr B6 – manipulate polynomials algebraicallyD1 – binomial expansionLinks to Chapter 13 Probability and statistical distributions: 13.1 The binomial distribution1.2Ex 1.2AEx 1.2B3 The binomial expansion Students are introduced to the binomial expansion formula as a way of solving any binomial expansion, they should state which formula they are using at the start of any answer.Understand and use the binomial expansion of (a + b)nB6 – manipulate polynomials algebraicallyD1 – binomial expansionThe binomial expansion is used in Chapter 13 Probability and statistical distributions for binomial distributions1.3Ex 1.3A4 Factorisation Students factorise different sorts of expressions including quadratics. They also factorise a difference of two squares. Some questions are given in relation to real life problems to show how factorisation is used.Manipulate polynomials algebraically including:factorisationB6 – manipulate polynomials algebraicallyFactorisation is used in Chapter 2 Algebra and functions 2: Equations and inequalities when working with quadratic equations and in Chapter 3 Algebra and functions 3: Sketching curves1.4Ex 1.4A5 Algebraic division All the skills are then combined to write algebraic fractions in their simplest form. Students work through the factor theorem. Students then divide polynomials by an expression – make it clear to students that they use the same method as they do for numerical long division. Manipulate polynomials algebraically including:expanding brackets and simplifying by collecting like termsfactorisationsimple algebraic divisionuse of the factor theoremB6 – manipulate polynomials algebraically, including expanding brackets, collecting like terms, factorisation and simple algebraic division; use of the factor theoremLink algebraic division to numerical division1.5Ex 1.5A6 Laws of indices, manipulating surdsStudents solve questions using the laws of indices, including negative and fractional indexes. They are reminded that surds are roots of rational numbers and simplify expressions involving surds. Understand and use the laws of indices for all rational exponents Use and manipulate surds A1 – mathematical proofB1 – understand and use the laws of indices for all rational exponentsB2 - use and manipulate surds, including rationalising the denominatorBuilds on GCSE skillsLaws of indices is used in Chapter 7 Exponentials and logarithms, section 7.2Manipulation of indices and surds is used in Chapter 8 Differentiation and Chapter 9 Integration1.61.7Ex 1.6AEx 1.7A7 Rationalising the denominatorRecap manipulating surds and complete Ex 1.7A if required and then students go on to rationalise the denominator of algebraic fractions.Use and manipulate surds, including rationalising the denominatorA1 – mathematical proofB2 - use and manipulate surds, including rationalising the denominator1.8Ex 1.8A8 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 1 in Student BookExam-style extension questions 1 in Student BookEnd-of-chapter test 1 on Collins ConnectCHAPTER 2 – Algebra and functions 2: Equations and inequalities (8 Hours) Prior knowledge neededChapter 1TechnologyUse a calculator to check solutions to quadratic equations that have been solved by hand.Use a graphing software to plot lines when solving simultaneous equations so that students can see that where the lines intersect is the solution.Use graphing software to show the solutions to quadratic inequalities.One hour lessonsLearning objectivesSubject content referencesTopic linksStudent Book references1 Quadratic functions Students draw and work out key aspects of quadratic functions, they also work out equations of given quadratic graphs. Make students aware that the notation f(x) may be used.Consider the difference between drawing and sketching.Work with quadratic functions and their graphsB3 – work with quadratics functions and their graphsMake connections back to factorising in 1.4 Factorisation2.1Ex 2.1A2 The discriminant of a quadratic function Students are reminded that the discriminant is b2–4ac and that the value of the discriminant indicates what type of roots and number of solutions a quadratic has. Link this to how students will sketch quadratics in the next chapter. Students need to make sure they have a quadratic in the form ax2+bx+c=0 before they calculate the discriminant.Calculate and use the discriminant of a quadratic function, including the conditions for real and repeated rootsB3 – work with the discriminant of a quadratic function, including the conditions for real and repeated rootsLinks to 3.1 Sketching curves of quadratic functions 2.2Ex 2.2A3 Completing the squareStudents complete the square on the general expression of a quadratic. They go onto solving quadratic equations by completing the square.The general formula for solving a quadratic equation can be derived by completing the square on the general form y=ax2+bx+c – this is question 6 in Ex 2.plete the squareA1 – mathematical proofB3 – complete the square5.1 Equations of circles uses completing the square when solving problems involving circlesLinks to 3.1 sketching curves of quadratic functions2.3Ex 2.3A4 Solving quadratic equations Students use factorisation, using the formula, completing the square and calculators to solve quadratic equations, they consider advantages and disadvantages of each method.It’s important not to be seen as a discrete set of skills but providing different information about the same equation.Solve quadratic equations using completing the squarethe formulafactorisationA1 – mathematical proofB3 – solve quadratic equationsThe ability to find solutions will help you solve distance, speed and time problems in Chapter 15 Kinematics and in Chapter 6 Trigonometry and Chapter 7 Exponentials and logarithmsLink to graph drawing, 3.1 Sketching curves of quadratic functions2.4Ex 2.4A5 Solving simultaneous equations Students go through solving two linear simultaneous equations by elimination and then by substitution. They need to think about which is the best method and why. Solve simultaneous equations in two variables by elimination or by substitution, including one linear and one quadratic equationB4 – solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equationThis is used when finding points of intersection with circles and straight lines in Chapter 5 Coordinate geometry 2: Circles2.5Ex 2.5AEx 2.5B6 Solving linear and quadratic simultaneous equations Student solve linear and quadratic simultaneous equations considering which method is the most appropriate to use.Solve simultaneous equations in two variables by elimination or by substitution, including one linear and one quadratic equationB4 – solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation2.6Ex 2.6A7 Solving linear and quadratic inequalitiesStudents solve inequalities, use set notation to list integer members of two or more combined inequalities and represent inequalities graphically. Strict inequalities are represented with a dotted line, inclusive inequalities are represented with a solid line.Link to solving equations, drawing graphs and looking at inequalities from the sketch of a graph.Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractionsExpress solutions through correct use of ‘and’ and ‘or’, or through set notationB5 - solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractionsB5 - express solutions through correct use of ‘and’ and ‘or’, or through set notationThis is used in Chapter 5 Coordinate geometry 2: Circles when solving problems involving equations of circles2.7Ex 2.7A2.8Ex 2.8A8 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 2 in Student BookExam-style extension questions 2 in Student BookEnd-of-chapter test 2 on Collins ConnectCHAPTER 3 – Algebra and functions 3: Sketching curves (9 Hours) Prior knowledge neededChapter 2TechnologyGraphing software can be heavily used in this chapter; some examples are below.Use graphing software to check answers.Use graphing software to zoom into a point of inflection and see what happens to the gradient of the curve each side of the point of inflection.Use graphing software to compare how to graphs behave.Use graphing software to look at approximate coordinates of turning points.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 Sketching curves of quadratic functionsStudents sketch curves of quadratic equations and find quadratic equations from sketches knowing that a sketch shows the curve along with its key features. The key features being; maximum or minimum turning point and its coordinates, the roots with x intercept/s if appropriate, the y intercept.Use and understand the graphs of functionsSketch curves defined by simple quadratic equationsB3 – work with quadratic functions and their graphsB7 – understand and use graphs of functions; sketch curves defined by simple equationsBuilds on from Chapter 2 Algebra and functions 2: Equations and inequalities3.1Ex 3.1A2 Sketching curves of cubic functionsStudents consider how the cubic functions behaves as it tends to positive and negative infinity, this determines the shape of the curve. The coordinates of the axes intercepts and point of inflection are included on the graph. Students use this information to sketch curves of cubic functions.Use and understand the graphs of functionsSketch curves defined by simple cubic equationsB7 - understand and use graphs of functions; sketch curves defined by simple equationsB7 – interpret algebraic solutions of equations graphicallyBuilds on from Chapter 2 Algebra and functions 2: Equations and inequalities3.2Ex 3.2AEx 3.2B3 Sketching curves of quartic functionsStudents find turning points and points of intersection to sketch curves of quartic functions. They look at maximum and minimum turning points and then at the roots of the equation. A typical example of a quartic to be sketched is y=x2(2x-1)2 .Use and understand the graphs of functionsSketch curves defined by simple quartic equationsInterpret the algebraic solution of equations graphicallyB7 - understand and use graphs of functions; sketch curves defined by simple equations B7 – interpret algebraic solutions of equations graphicallyBuilds on from Chapter 2 Algebra and function 2: Equations and inequalities3.3Ex 3.3AEx 3.3B4 Sketching curves of reciprocal functionsStudents sketch graphs of reciprocals of linear y=ax and quadratic y=ax2 functions. They are introduced to the word discontinuity and know that this is a point where the value of x or y is not defined. They also are told that an asymptote is a straight line that a curve approaches but never meets, these should be clearly highlighted on the graph. They also give the points of interception on the axis.Use and understand the graphs of functionsSketch curves defined by simple reciprocalsInterpret the algebraic solution of equations graphicallyB7 - understand and use graphs of functions; sketch curves defined by simple equations including polynomials y=ax and y=ax2 (including their vertical and horizontal asymptotes)Graph sketching skills are further developed in Chapter 7 Exponentials and logarithms3.4Ex 3.4AEx 3.4B5 Intersection pointsStudents sketch more than one function on a graph to work out the intersection points of the functions and link this to solve an equation which includes these functions. They use the information they have learnt for the previous section in this exercise to solve these problems.Use intersection points of graphs to solve equationsB7 interpret algebraic solution of equations graphically; use intersection points of graphs to solve problemsLinks back to 2.5 Solving simultaneous equations 3.5Ex 3.5A6 Proportional relationshipsStudents express the relationship between two variables using the proportion symbol and then find the equation between the two variables. They also recognise and use graphs that show proportion.Understand and use proportional relationships and their graphsB7 – understand and use proportional relationships and their graphsLinks back to GCSE3.6Ex 3.6A7 Translations Students transform functions and sketch the resulting graphs using the two transformations identified in the learning objective. Understand the effects of simple transformations on the graphs of y=f(x), including y=f(x)+a and y=f(x+a) B9 – effects of simple transformations including sketching associated graphsTranslations and stretches arise in 6.5 solving trigonometric equations3.7Ex 3.7A8 StretchesStudents transform functions and sketch the resulting graphs using the two transformations identified in the learning objective.Understand the effects of simple transformations on the graphs of y=f(x), including y=af(x) andy=f(ax).B9 – effects of simple transformations including sketching associated graphsTranslations and stretches arise in 6.5 solving trigonometric equations3.8Ex 3.8A9 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 3 in Student BookExam-style extension questions 3 in Student BookEnd-of-chapter test 3 on Collins ConnectCHAPTER 4 – Coordinate geometry 1: Equations of straight lines (5 Hours)Prior knowledge neededChapter 1TechnologyUse graphing software to check answers.Use graphing software to investigate the location of the x-intercept as the gradient varies with both positive and negative and integer and fractional values.Use graphing software to find equations of lines that are parallel.Use graphing software to plot lines that intersect but are not perpendicular. What do the students notice about the products of their gradients?One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 Straight line graphs – part 1Students rearrange equations of straight lines to the form ax+by+c=0Students find the equation of a line given the gradient and a point on the line and solve problems using this.Write the equation of a straight line in the form ax+by+c=0 Find the equation of a straight line through one point with a given gradient using the formula y–y1=m(x–x1) C1 – understand and use the equation of a straight line, including the formsy–y1=m(x–x1) and ax+by+c=04.2 and 4.3 link to Chapter 15 Kinematics4.1Ex 4.1A4.2Ex 4.2A2 Straight line graphs – part 2Students find the gradient given two points on the line and solve problems using this.Given two points on a line students use the formula to find the equation of the straight line.Find the gradient of the straight line between two points Find the equation of a straight line using the formula y–y1y2–y1=x–x1x2–x1C1 – understand and use the equation of a straight line, including the forms y–y1=m(x–x1) and ax+by+c=0Graph sketching skills are further developed in Chapter 7 Exponentials and logarithms4.3Ex 4.3A4.4Ex 4.4A3 Parallel and perpendicular linesStudents use the facts about gradients of parallel and perpendicular straight lines to solve problems.Understand and use the gradient conditions for parallel linesUnderstand and use the gradient conditions for perpendicular linesC1 – gradient conditions for two straight lines to be parallel or perpendicularLinks back to GCSE4.5Ex 4.5AEx 4.5B4 Straight line modelsStudents apply what they have learnt in this chapter to solve problems in a variety of different contexts including real-world scenarios.Be able to use straight line models in a variety of contextsC1 – be able to use straight line models in a variety of contextsThis is used in Chapter 5 Coordinate geometry 2: Circles when finding a gradient to solve problems involving tangents and radii of circles and points of intersections of circles with straight lines4.6Ex 4.6A5 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 4 in Student BookExam-style extension questions 4 in Student BookEnd-of-chapter test 4 on Collins ConnectCHAPTER 5 – Coordinate geometry 2: Circles (6 Hours)Prior knowledge neededChapter 2Chapter 4TechnologyUse graphing software draw x2+y2=25 and x2+y2=36 to show how the centre of a circle is always (0, 0) and the radius is the square root of the number on the right-hand side of the equation. Get students to predict what other circles will look like given an equation of a circle in this form.Graphing software can be used to look at other circles (still with the equation written in this form but not with centre (0, 0) ) and link the equation of the circle to the coordinates of the centre and length of radius.Use graphing software to find midpoints of line segments.Use graphing software to plot a circle and tangent.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 Equations of circles – part 1 Students link equations of circles to Pythagoras’ theorem. They learn the general equation of a circle and solve problems. They find the equation of a circle from its centre and a point on the circumference. Given an equation of a circle students find the centre and radius by completing the square on x and y.Find the centre of the circle and the radius from the equation of a circleWrite the equation of a circleC2 – understand and use the coordinate geometry of the circle including using the equation of a circle in the form (x–a)2+(y–b)2=r2C2 – complete the square to find the centre and radius of a circleLinks back to GCSE5.1Ex 5.1A2 Equations of circles – part 2Recap lesson one and then students find the equation of a circle from the ends of a diameter, this involves them using the formula for the midpoint of a line segment.Find the centre of the circle and the radius from the equation of a circleWrite the equation of a circleC2 – understand and use the coordinate geometry of the circle including using the equation of a circle in the form (x–a)2+(y–b)2=r2C2 – complete the square to find the centre and radius of a circle5.1Ex 5.1B3 Angles in semicirclesA geometric proof is used to show that for three points on the circumference of a circle, if two points are on a diameter, then the triangle is right-angled. Students are also reminded that Pythagoras’ theorem will show that a triangle has a right angle and gradients can be used to show that two lines are perpendicular.Find the equation of a circle given different information such as the end points of a diameter Use circle theorems to solve problemsA1 - mathematical proofC2 - use of the following properties:the angle in a semicircle is a right angleLinks back to GCSE5.2Ex 5.2A4 Perpendicular from the centre to a chordA geometric proof is used to show that congruent triangles are identical and then links to show that the perpendicular from the centre of the circle to a chord bisects the chord.Find the equation of a perpendicular bisectorUse circle theorems to solve problemsC2 - use of the following properties: the perpendicular from the centre to a chord bisects to chordLinks back to GCSE5.3Ex 5.3A5 Radius perpendicular to the tangentA geometric proof by contradiction is used to show that a radius and tangent are perpendicular at the point of contact. Pythagoras’ theorem is used to find the length of a tangent. Students then go on to consider that for any circle there are two tangents with the same gradient. Given the gradient of the tangent and the equation of the circle they can find both points where the two tangents touch the circle. The second method used it to show that a line is tangent to a circle is that the discriminant will be zero i.e. b2–4ac=0.Find the equation and length of a tangentUse circle theorems to solve problemsC2 - use of the following properties: the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point5.4Ex 5.4AEx 5.4B6 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 5 in Student BookExam-style extension questions 5 in Student BookEnd-of-chapter test 5 on Collins ConnectCHAPTER 6 – Trigonometry (7 Hours)Prior knowledge neededChapter 2Chapter 3TechnologyUse calculators to check values.Use graphic software to confirm the shapes of graphs.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 Sine and cosineStudents extend their knowledge of sine and cosine beyond 90° to between 0° and 180° by imagining a rod of length one unit on a coordinate grid.Define sine for any angleDefine cosine for any angleE1 – understand and use the definitions of sine and cosine for all argumentsLinks back to GCSEBearings will be extended further in Chapter 10 Vectors6.1Ex 6.1A2 The sine rule and the cosine ruleStudents work through proofs for the sine rule, cosine rule and formula for the area of a triangle and then use these to solve problems.Use the sine ruleUse the cosine ruleUse the formula for the area of a triangleE1 - the sine and cosine ruleE1 – the area of a triangle in the form 12absinCLinks back to GCSE6.2Ex 6.2A3 Trigonometric graphs Students start by considering the sine and cosine of any angle using x and y coordinates of points on the unit circle. This is then linked to the trigonometric graphs and their symmetries and periodicity.Understand the graph of the sine function and interpret its symmetry and periodicityUnderstand the graph of the cosine function and interpret its symmetry and periodicityE3 – Understand and use the sine and cosine functions; their graphs, symmetries and periodicityLink back to GCSE6.3Ex 6.3A4 The tangent functionStudents consider when the tangent cannot be calculated and look at the similarities between the tangent graph and the sine and cosine graphs. They use tan?=sin?cos? to solve problems.Define the tangent for any angleUnderstand the graph of the tangent function and interpret its symmetry and periodicityUse an important formula connecting sin x, cos x and tan xE1 – understand and use the definition of tangent for all argumentsE3 - Understand and use the tangent functions; its graph, symmetry and periodicityE5 – understand and use tan?=sin?cos? Link to Chapter 4 Coordinate geometry 1: Equations of straight lines6.4Ex 6.4A5 Solving trigonometric equationsStudents thing about the multiple solutions of trigonometric equations and link this to the graphs of the function. Solve simple trigonometric equationsE7 – solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angleLink this back to transformation of graphs in Chapter 3 Algebra and functions 36.5Ex 6.5A6 A useful formulaStudents diagrammatically see that sin2?+cos2?=1 and then use this identity to go on to solve trigonometric equations.Link to Pythagoras’ theorem.Use an important formula connecting sin x, cos x and tan xE5 – understand and use sin2?+cos2?=1Link to Pythagoras’ theorem.6.6Ex 6.6A7 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 6 in Student BookExam-style extension questions 6 in Student BookEnd-of-chapter test 6 on Collins ConnectCHAPTER 7 – Exponentials and logarithms (8 Hours)Prior knowledge neededChapter 1Chapter 2Chapter 3Chapter 4TechnologyGraph plotting software to draw the graphs y = 1.2x.Use the log, power of e and ln button on a calculator.Use spreadsheet software to calculate values of logarithms.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 The function axStudents learn the terms exponential growth and exponential decay. The draw graphs of the form y=ax and are made aware that there is a difference in the shape of the graph when a < 1 and a > 1, they should understand this.Use and manipulate expressions in the form y=ax where the variable x is an indexDraw and interpret graphs of the form y=axUse the graph to find approximate solutionsF1 – Know and use the function ax and its graph, where a is positiveLinks back to GCSE7.1Ex 7.1A2 LogarithmsStudents start with the laws of logarithms in base 10 and the laws of logarithms are linked to laws of indices with a proof for each law. They work out basic logarithms in base 10 without a calculator. They then move onto using the laws for logarithms of any base and using the logab button on a calculator.Understand and use logarithms and their connection with indicesUnderstand the proof of the laws of logarithmsF3 – know and use the definition of logax as the inverse of ax, where a is positive and x≥0F4 – understand and use the laws of logarithmsThe skills learnt in 1.6 Laws of indices are used here7.2Ex 7.2AEx 7.2B3 The equation ax=bStudents solve equations of the form ax=b by taking the logarithms of both sides. The work through a proof of the formula to change the base of a logarithm.Solve equations in the form ax=bUnderstand a proof of the formula to change the base of a logarithmF5 – solve equations in the form ax=b7.3Ex 7.3A4 Logarithmic graphsStudents look at two different types of curve y=axn and y=kbx. To find the parameters they take the logarithms of both sides and then rearrange this equation into the form y=mx+c. In the first case they plot log y against log x and obtain a straight line where the intercept is log a and the gradient is n and in the second case they plot log y against x and obtain a straight line where the intercept is log k and the gradient is log b.Use and manipulate expressions in the form y=kax where the variable x is an indexUse logarithmic graphs to estimate parameters in relationships of the form y=axn and y=kbx, given data for x and yF3 – know and use the definition of logax as the inverse of ax, where a is positive and x≥0F6 – use logarithmic graphs to estimate parameters in relationships of the form y=axn and y=kbx, given data for x and y7.4Ex 7.4A5 The number eStudents look at graphs of the form y=ax . and see the gradient is always a multiplier x y. When the multiplier is one then the function is of the form y=ex and go on to generalise that for y=eax then the gradient = aeax = ay for any value of a. Students should realise that when the rate of change is proportional to the y value, an exponential model should be used. Insure that students include the graphs of y=eax+b+c.Know that the gradient of ekx is equal to kekx and hence understand why the exponential model is suitable in many applicationsF1 – Know and use the function ex and its graphF2 – Know that the gradient of ekx is equal to kekx and hence understand why the exponential model is suitable in many applications7.5Ex 7.5A6 Natural logarithmsStudents are introduced to the fact that lny means logarithm to the base e and that if ex=y then lny=x. The graphs of y=lnx and y=ex show students that they are a reflection of each other in the line y=x due to the fact they are inverse function. Solutions of equations of the form eax+b=p and ln(ax+b)=q is expected.Know and use the function lnx and its graphsKnow and use lnx as the inverse function of exF3 – know and use the function lnx and its graphF3 – know and use lnx as the inverse function of ex7.6Ex 7.6A7 Exponential growth and decayEquations of exponential growth and decay are often written in terms of powers of e as it makes it easy to find the rate of change at any time. Exponential growth and decay are often described by a relationship of the form y=aekx where a and k are parameters. For growth a>0 and for decay a<0.Recognise and interpret exponential growth and decay in modelling changeF7 – understand and use exponential growth and decay; use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models7.7Ex 7.7A8 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 7 in Student BookExam-style extension questions 7 in Student BookEnd-of-chapter test 7 on Collins ConnectCHAPTER 8 – Differentiation (8 Hours)Prior knowledge neededChapter 1Chapter 3Chapter 4TechnologyUse graphical software to find the gradients of a quadratic curve in example 2 in section 8.2.Graphical software can be used to look for stationary points – students also need to find them using differentiation.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 The gradient of a curveStudents are introduced to the concept that the gradient of a curve at any point is equal to the gradient of the tangent at that point. Students then look at y=x2 and see that the gradient at any point on the curve is 2x. The derivative is then introduced as the connection between the x coordinate of a point and the gradient of the curve at that point.To consider the relationship between the tangent to a curve and the gradient of the curve at a pointIntroduction of the term derivativeRecognise the derivative function of y=x2G1 – understand and use the derivative of f(x) as the gradient of the tangent to a graph of y=f(x) at a general point (x,y); the gradient of the tangent as a limit; interpretation as a rate of changeThe skills learnt in this chapter are essential when students learn Chapter 9 Integration8.1Ex 8.1A8.2Ex 8.2A2 The gradient of a quadratic curveFrom exercise 8.2A students are lead to the general result that if y=ax2 then dydx=2ax. After some examples students are given the general result when differentiating a quadratic. It is also made clear that terms should be differentiated separately in a quadratic equation. They start to sketch curves of the derivative.Extend the result for the derivative of y=x2 to the general case y=ax2 Be able to differentiate any quadratic equation G1 – sketching the gradient function for a given curveG2 – differentiate xn, for rational values of n, and related constant multiples, sums and differences8.2Ex 8.2B3 Differentiation of x2 and x3Students work through a proof that the derivative of x2 is 2x, they are told that using the formula is called differentiation from first principles.Understand a proof that if f(x)=x2 then f’(x)=2xG1 – differentiation from first principals for small positive integer powers of x G2 - differentiate xn, for rational values of n, and related constant multiples, sums and differences8.3Ex 8.3A4 Differentiation of a polynomialThe general result for differentiation is introduced to students and examples are worked through. They then go on to looking at differentiating the sum or difference of terms.Differentiate expressions containing integer powersG2 - differentiate xn, for rational values of n , and related constant multiples, sums and differences8.4Ex 8.4A5 Differentiation of xnStudents move on to work with powers that are not integer values and variables other than x and y. Differentiation is again linked to being the gradient of the graph and the rate of change.Differentiate expressions containing powers of variablesG2 - differentiate xn, for rational values of n, and related constant multiples, sums and differences8.5Ex 8.5AEx 8.5B6 Stationary points and the second derivativeDifferentiation is used to find stationary points, i.e. the point where dydx=0. Students are shown that the second derivative will show if a turning point is a maximum or minimum point so there is no need to draw the graph to ascertain this information.Determine geometrical properties of a curveG1 – second derivativesG1 – understand and use the second derivative as the rate of change of gradientG3 – apply differentiation to find gradients, tangents and normal, maxima and minima and stationary pointsG3 – identify where functions are increasing and decreasing8.6Ex 8.6A7 Tangents and normalThe normal at any point on a curve is perpendicular to the tangent at that point. As the product of their gradients is -1 differentiation can be used to find the equations of the tangent and the normal.Find equations of tangents and normal using differentiationG3 – apply differentiation to find gradients, tangents and normal, maxima and minima and stationary pointsG3 – identify where functions are increasing and decreasing8.7Ex 8.7A8 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 8 in Student BookExam-style extension questions 8 in Student BookEnd-of-chapter test 8 on Collins ConnectCHAPTER 9 – Integration (5 Hours)Prior knowledge neededChapter 1Chapter 3Chapter 8TechnologyGraphics calculators/software can be used to look at areas under a curve and model changes.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 Indefinite integralsStudents look at integration being the reverse process of differentiation. Because the derivative of the final constant is zero the result of integration is sometimes called the indefinite integral because the value of c is not fixed. Students integrate the sum or difference of terms by integrating them separately.Make use of the connections between differentiation and integrationH1 – Know and use the Fundamental Theorem of CalculusH2 – Integrate xn (excluding n=-1), and related sums, differences and constant multiplesThe fundamental theorem of calculus is used in 15.4 Displacement-time and velocity-time graphs9.1Ex 9.1A2 Indefinite integrals continuedStudents move the learning from the first lesson forward by finding the equations of the curve given the derivative and a point on the curveFind the equation of a curve when you know the gradient and point on a curveH1 – Know and use the Fundamental Theorem of CalculusH2 – Integrate xn (excluding n=-1), and related sums, differences and constant multiples9.1Ex 9.1B3 The area under a curveStudents move on to calculate definitive integrals and are introduced to the notation. This enables them to calculate the area under curves. Students complete the first 6 questions of exercise 9.2AUse a technique called integration to find the area under a curveH2 – Integrate xn (excluding n=-1), and related sums, differences and constant multiplesH3 – Evaluate definite integrals; use a definite integral to find the area under a curve9.2Ex 9.2A4 The area under a curve continuedRecap knowledge from this and the previous chapter (differentiation) ensuring students clearly understand the link between differentiation and integration. Students finish exercise 9.2A.Use a technique called integration to find the area under a curveH2 – Integrate xn (excluding n=-1), and related sums, differences and constant multiplesH3 – Evaluate definite integrals; use a definite integral to find the area under a curve9.2Ex 9.2A5 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 9 in Student BookExam-style extension questions 9 in Student BookEnd-of-chapter test 9 on Collins ConnectCHAPTER 10 – Vectors (5 Hours)Prior knowledge neededChapter 6TechnologyUse a graphics package to plot vectors and investigate how the angle changes depending on the orientation.Plotting diagrams using graphics software enables you to visualise the situation and see how the vectors will still satisfy the rules even if the orientation of the vectors is not identical.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 Definition of a vectorStudents look at vector notation and use Pythagoras’ theorem to calculate the magnitude of vectors. They use trigonometry to find the direct of a vector or use bearings. They calculate the unit vector and use this to find a velocity vector.Write a vector in i j or as a column vectorCalculate the magnitude and direction of a vector as a bearing in two dimensionsIdentify a unit vector and scale it up as appropriateFind a velocity vectorJ1 – use vectors in two dimensionsJ2 – calculate the magnitude and direction of a vector and convert between component form and magnitude/direction formJ1 – use vectors in two dimensionsJ2 – calculate the magnitude and direction of a vector and convert between component form and magnitude/direction formTrigonometric bearings were introduced in 6.2 The sine rule and the cosine rule10.1Ex 10.1A2 Adding vectorsStudents add and subtract vectors in both i j and column notation. They are introduced to the three types of vectors (equal, opposite and parallel) and learn that parallel vectors contain a common factor.Add vectorsMultiply a vector by a scalarJ3 – add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalarsJ3 – add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalarsAdding vectors is used to find resultant forces in Chapter 16 Forces10.2Ex 10.2A3 Vector geometryStudents add vectors using the triangle law and are introduced to the parallelogram law which shows that that the order in which vectors are added is not important. Students go on to be introduced to the ratio theorem and the midpoint rule (a special case of the midpoint theorem)Apply the triangle law to add vectorsApply the ratio and midpoint theoremsJ3 – add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars and understand their geometric interpretationsJ3 – add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars and understand their geometric interpretations10.3Ex 10.3AEx 10.3B4 Position vectorsStudents learn that the position vector of a point is its position relative to the origin. They use this to calculate the relative displacement between two position vectors by subtracting the position vectors. Students calculate the distance between the two points by finding the magnitude of the relative displacement using Pythagoras’ theorem.Find the position vector of a pointFind the relative displacement between two position vectorsJ4 – understand and use position vectors; calculate the distance between two points represented by position vectorsJ4 – understand and use position vectors; calculate the distance between two points represented by position vectors10.4Ex 10.4A5 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 10 in Student BookExam-style extension questions 10 in Student BookEnd-of-chapter test 10 on Collins ConnectCHAPTER 11 – Proof (4 Hours)Prior knowledge neededProofs in context are flagged with boxes in every chapter. It is recommended to teach this chapter as a summation of proof and to ensure tested skills have been mastered.Students will need well developed algebra skills to write and understand proofs.TechnologyA spreadsheet package can be used to generate values.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 Proof by deductionStudents are reminded that proof by deduction was used when proving the sine and cosine rules in Chapter 6 Trigonometry. They work through examples where conclusions are drawn by using general rules in mathematics and working through logical steps.Students should be made aware that this is the most commonly used method of proof throughout this specificationUse proof by deductionA1 – understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusionA1 – use methods of proof; including proof by deductionNote: Proofs in context are flagged with boxes in every chapter. It is recommended to teach this chapter as a summation of proof and to ensure tested skills have been mastered11.1Ex 11.1A2 Proof by exhaustionStudents learn that when they complete a proof by exhaustion they need to divide the statement into a finite number of cases, prove that the list of cases identified is exhaustive and then prove that the statement is true for all the cases. I.e. they need to try all the options Use proof by exhaustionWork systematically and list all possibilities in a structured and ordered wayA1 - understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusionA1 – use methods of proof; including proof by exhaustion11.2Ex 11.2A3 Disproof by counter exampleStudents need to show that a statement is false by finding one exception, a counter exampleUse proof by counter exampleA1 - understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusionA1 – use methods of proof; including proof by counter example11.3Ex 11.3A4 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 11 in Student BookExam-style extension questions 11 in Student BookEnd-of-chapter test 11 on Collins ConnectCHAPTER 12 – Data presentation and interpretation (11 Hours)Prior knowledge neededChapter 1Chapter 4TechnologyCalculators can be used to calculate many useful statistics – put the calculator into statistics mode.Use frequency tables on calculators.Use a calculator to calculate the mean and standard deviation using the frequency table option.Use graph-drawing software to plot cumulative frequencies.Use graphing software to create histograms.Use spreadsheet packages for the data set activities.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 Measures of central tendencyStudents review types of data, the median, the mode and the mean. Students consider which is the best measure of central tendency to use and why. They use Data set activity 12.1 to calculate the median and mean and comment critically on findings.Use discrete or continuous dataUse measures of central tendency: mean, median, modeL3 – interpret measure of central tendency and variationLinks back to GCSE12.1DS 12.12 Measures of central tendency and spread Students review measures of spread; the range and interquartile rangeThey use frequency table to calculate measure of central tendency and spread and estimate the mean from frequency tables.Use discrete, continuous, grouped or ungrouped dataUse measures of variation: range and interpercentile rangesL3 – interpret measure of central tendency and variationLinks back to GCSE12.1Ex 12.1A3 Variance and standard deviation Students are introduced to variance and standard deviation and the fact that these measures use all the data items. Students go onto calculating variance and standard deviation using frequency tables. Use measure of variation: variance and standard deviationUse the statistic sxx=x-x2=x2x2nUse standard deviation σ=SxxnL3 – interpret measures of variation, extending to standard deviationL3 – be able to calculate standard deviation, including from summary statisticsL4 – select or critique data presentation techniques in the context of a statistical problemThis is covered further in Book 2, Chapter 13 Statistical distributions12.2Ex 12.2A4 Data cleaning Recap calculating variance and standard deviation from frequency tables and then introduce data cleaning as involving the identification and removal of any invalid data points from the data set. Students use Data set activity 12.2 to clean the data and give reasons for their omissions. Use measure of variation: variance and standard deviationUse the statistic sxx=x-x2=x2x2nUse standard deviation σ=SxxnL4 – select or critique data presentation techniques in the context of a statistical problem12.2DS 12.25 Standard deviation and outliers Student look at data sets that contain extreme values, if a value is more than two standard deviations away from the mean it should be treated as an outlier. Students use Data set activity 12.3 to apply what they have learnt so far, calculating the mean and standard deviation, and cleaning a section of the large data set.Use measure of variation: variance and standard deviationUse the statistic sxx=x-x2=x2x2nUse standard deviation σ=SxxnL3 – interpret measures of variation, extending to standard deviationL3 – be able to calculate standard deviation, including from summary statisticsL4 – select or critique data presentation techniques in the context of a statistical problemNormal distributions are covered in more depth in Book 2, Chapter 13 Statistical distributions12.2DS 12.36 Linear coding Students then use linear coding to make certain calculations easier. They apply a linear code to a section of large data in Data set activity 12.4.Learning from this and the previous three lessons is consolidated in exercise 12.2BUnderstand and use linear codingL3 – interpret measures of variation, extending to standard deviationL3 – be able to calculate standard deviation, including from summary statisticsL4 – select or critique data presentation techniques in the context of a statistical problem12.2DS 12.4Ex 12.2B7 Displaying and interpreting data – box and whisker plots and cumulative frequency diagramsBox and whiskers plots are used to compare sets of data. The way of showing outliers on a box and whisker plot and identifying outliers using quartiles is explained. Cumulative frequency diagrams are used to plot grouped data. (0, 0) should be plotted and the cumulative frequencies should be plotted at the upper bound of the interval. Percentiles are estimated from grouped data. Interpolation is used to find values between two points.Interpret and draw frequency polygons, box and whisker plots (including outliers) and cumulative frequency diagramsUse linear interpolation to calculate percentiles from grouped dataL1 – Interpret diagrams for single-variable data, including understanding that area in a histogram represent frequencyL1 – connect probability to distributions12.3Ex 12.3A8 Displaying and interpreting data – histogramsStudents plot histograms and look at the distribution of the data. They apply what they have learnt to Data set activity 12.5, creating a histogram and performing linear interpolation.Interpret and draw histogramsL1 – Interpret diagrams for single-variable data, including understanding that area in a histogram represent frequency12.3DS 12.5Ex 12.3B9 Displaying and interpreting data – scatter graphs Students plot scatter diagrams with bivariate data and look for a correlation in the data. Students are introduced to the concept of spurious correlation.In Data set activity 12.6 students display a selection of data in a scatter graph and use their data interpretation skills to identify outliers and comment on trends. Describe correlation using terms such as positive, negative, zero, strong and weakL2 – understand informal interpretation of correlationL2 – interpret scatter diagrams and regression lines for bivariate data, including recognition of scatter diagrams which include distinct sections of the population (calculations involving regression lines are excluded)L2 – understand that correlation does not imply causation12.3DS 12.6Ex 12.3C10 Displaying and interpreting data – regression lines, lines of best fit, interpolation and extrapolationHaving plotted scatter diagrams in the previous lesson students go on to look at the link between the line of best fit and regression line, y=a+bx where b is the gradient and a is the y-intercept. Equations of regression lines are then explored. This is then used for interpolation and extrapolation to make predictions. Finally, students look at the correlation between two variables in Data set activity 12.7.Use and interpret the equation of a regression line using statistical functions on your calculator and make predictions within the range of values of the explanatory variable and understand the dangers of extrapolationL2 – interpret scatter diagrams and regression lines for bivariate data, including recognition of scatter diagrams which include distinct sections of the population (calculations involving regression lines are excluded)L2 – understand that correlation does not imply causationGradients and intercepts were covered in Chapter 4 Coordinate geometry 1: Equations of straight lines12.3DS 12.7Ex 12.3D11 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 12 in Student BookExam-style extension questions 12 in Student BookEnd-of-chapter test 12 on Collins ConnectCHAPTER 13 – Probability and statistical distributions (5 Hours)Prior knowledge neededChapter 1Chapter 12TechnologyUse a calculator to perform exact binomial distribution calculations.Use a calculator to perform cumulative binomial distribution calculations.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 Calculating and representing probabilityBasic probability is recapped and sample space diagrams are used to determine the probability of equally likely events happening. Students go on to calculate probabilities of mutually exclusive and independent events using Venn diagrams and tree diagrams.Use both Venn diagrams and tree diagrams to calculate probabilitiesM1 – understand and use mutually exclusive and independent events when calculating probabilityLinks back to GCSEThis will help you in Book 2, Chapter 12 Probability13.1Ex 13.1A2 Discrete and continuous distributions – part 1The term discrete and continuous data is recapped. A discrete random variable is then defined. Probability distribution tables are used to show all the possible values a discrete random variable can take along with the probability of each value occurring. When all events are equally likely they form a uniform distribution. Students go on to probability distributions of a random variable. Identify a discrete uniform distributionM1 – link to discrete and continuous distributionsYou will use knowledge from Chapter 12 Data presentation and interpretation13.2Ex 13.2A3 Discrete and continuous distributions – part 2Continuous random variables are considered and when graphed a continuous line is used. The graph is called a probability density function (PDF), the area under a pdf shows the probability and sums to 1. Students then go on to find cumulative probabilities.Identify a discrete uniform distributionM1 – link to discrete and continuous distributionsYou will use knowledge from Chapter 12 Data presentation and interpretation13.2Ex 13.2B4 The binomial distributionBinomial distribution is about ‘success’ and ‘failure’ as there are only two possible outcomes to each trail. Students identify situations which can be modelled using the binomial distribution. They then calculate probabilities using the binomial distribution. Several examples are worked through.Use a calculator to find individual or cumulative binomial probabilities Calculate binomial probabilities using the notation X~B(n,p)N1 – understand and use simple, discrete probability distributions (calculation of mean and variance of discrete random variables excluded), including the binomial distribution, as a model; calculate probabilities using the binomial distributionYou will need the skills and techniques learnt in Chapter 1 Algebra and functions 1: Manipulating algebraic expressions Knowing how to calculate binomial probabilities will help you in Chapter 14 Statistical sampling and hypothesis testing13.3Ex 13.3A5 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 13 in Student BookExam-style extension questions 13 in Student BookEnd-of-chapter test 13 on Collins ConnectCHAPTER 14 – Statistical sampling and hypothesis testing (6 Hours)Prior knowledge neededChapter 12Chapter 13TechnologyUse a calculator to calculate binomial cumulative probabilities.Use spreadsheet packages for the data set activities.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 Populations and samplesDiscuss the different terms and terminology within populations including; finite and infinite population, census, sample survey, sampling unit, sampling frame and target population. Move onto looking at the different types of sampling; random sampling, stratified sampling, opportunity sampling, quota sampling, cluster sampling and systematic sampling. In Data set activity 14.1, students compare a random sample with a systematic sample from the large data set and draw conclusions.Use sampling techniques, including simple random, stratified sampling, systemic sampling, quota sampling and opportunity (or convenience) samplingSelect or critique sampling techniques in the context of solving a statistical problem, including understanding that different sample can lead to different conclusions about populationA1 – mathematical proofK1 – Use sampling techniques, including simple random, stratified sampling, systemic sampling, quota sampling and opportunity (or convenience) samplingK1 – Select or critique sampling techniques in the context of solving a statistical problem, including understanding that different sample can lead to different conclusions about populationYou will need to know how to clean data for Chapter 12 Data presentation and interpretation14.1DS 14.1Ex 14.1A2 Hypothesis testing – part 1The two different types of hypothesis are explained, the null and alternative hypotheses. 1-tail and 2-tail hypothesis are explained in relation to a six-sided die. Having carried out an experiment the null hypothesis can only be accepted or rejected based on the evidence, it cannot categorically be concluded from the findings.Students apply what they have learnt in this lesson in Data set activity 14.2.Apply the language of statistical hypothesis testing, developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-valueHave an informal appreciation that the expected value of a binomial distribution as given by np may be required for a 2-tail testConduct a statistical hypothesis test for the proportion in the binomial distribution and interpret that results in contextO1 – understand and apply the language of statistical hypothesis testing, develop through a binomial model: null hypothesis, alternative hypotheses, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-valueO2 – conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in contextYou will need to know Chapter 13 Probability and statistical distributions14.2DS 14.23 Hypothesis testing – part 2Recap previous lesson and students go on to complete exercise 14.2AApply the language of statistical hypothesis testing, developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-valueHave an informal appreciation that the expected value of a binomial distribution as given by np may be required for a 2-tail testConduct a statistical hypothesis test for the proportion in the binomial distribution and interpret that results in contextO1 – understand and apply statistical hypothesis testing, develop through a binomial model: null hypothesis, alternative hypotheses, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-valueO2 – conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context14.2Ex 14.2A4 Significance levels – part 1 Students look at significance levels being the probability of mistakenly rejecting the null hypothesis. The terms test statistic and critical region are explained and students move onto finding the critical region and critical value. They work through the examples up to the end of example 3.Appreciate that the significance level is the probability of incorrectly rejecting the null hypothesisO2 – understand that a sample is being used to make an inference about the population and appreciate that they significance level is the probability of incorrectly rejecting the null hypothesis14.25 Significance levels – part 2 Recap previous lesson, with a focus on how to calculate values using calculators instead of statistical tables. After working through the rest of the chapter from example 4 onwards, students go on to complete exercise 14.2B.Appreciate that the significance level is the probability of incorrectly rejecting the null hypothesisO2 – understand that a sample is being used to make an inference about the population and appreciate that they significance level is the probability of incorrectly rejecting the null hypothesis14.2Ex 14.2B6 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 14 in Student BookExam-style extension questions 14 in Student BookEnd-of-chapter test 14 on Collins ConnectCHAPTER 15 – Kinematics (10 Hours) Prior knowledge neededChapter 2Chapter 3Chapter 4Chapter 8Chapter 9Chapter 10TechnologyGraphics packages can be used to visualise problems.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 The language of kinematicsStudents are introduced to motion in a straight line and the vocabulary displacement, velocity and acceleration (vectors quantities) as opposed to distance, time and speed (scalar quantities). They use the SI unit for distance (metres) and time (seconds) to derive other units. Time permitting introduce the five SUVAT equations. Find the SI units of displacement, velocity and accelerationP1 – understand and use fundamental quantities and units in the S.I. system: length, time, massP1 – understand and use derived quantities and units; velocity and accelerationQ1 – understand and use the language of kinematics; position; displacement; distance travelled; velocity; speed; acceleration15.1Ex 15.1A2 Equations of constant acceleration – part 1The definition of acceleration (a=(v-u)t) is used to derive the five SUVAT equations. Students have to work out which of the equations to use to solve problems. Derive and use the equations of constant accelerationQ3 – understand, use and derive the formulae for constant acceleration for motion in a straight lineThe SUVAT equations are used in Chapter 16 Forces15.2Ex 15.2A3 Equations of constant acceleration – part 2Recap the five SUVAT equations from the previous lesson. Questions become more complicated in ex 15.2B and students are encouraged to draw a diagram to assist them in understanding the situationDerive and use the equations of constant accelerationQ3 – understand, use and derive the formulae for constant acceleration for motion in a straight line15.2Ex 15.2B4 Vertical motion – part 1Students continue to solve problems using the equations of constant acceleration but this time the acceleration is equal to gravity. They need to decide if gravity is influencing the situation in a positive or negative way. Derive and use the equations of constant acceleration, including for situation with gravityQ3 – understand, use and derive the formulae for constant acceleration for motion in a straight line 15.3Ex 15.3A5 Vertical motion – part 2Recap the previous lesson and move on to look at when two objects are launched in different ways, students are advised to look for what the journeys have in common.Derive and use the equations of constant acceleration, including for situation with gravityQ3 – understand, use and derive the formulae for constant acceleration for motion in a straight line 15.3Ex 15.3B6 Displacement-time and velocity-time graphs – part 1Where acceleration is constant, students construct graphs using straight lines and the concepts that the gradient of a displacement-time graphs is velocity and the gradient of a velocity-time graph is acceleration. The area under a velocity-time graph is used to calculate displacement. Students focus on displacement-time graphs.Draw and interpret displacement-time graphs and velocity-time graphsQ2 – understand, use and interpret graphs in kinematics for motion in a straight line; displacement against time and interpretation of gradient; velocity against time and interpretation of gradient and area under the graph15.4Ex 15.4A7 Displacement-time graphs and velocity-time graphs – part 2Recap the facts from last lesson and students go on to answer questions focusing on velocity-time graphs.Draw and interpret displacement-time graphs and velocity-time graphsQ2 – understand, use and interpret graphs in kinematics for motion in a straight line; displacement against time and interpretation of gradient; velocity against time and interpretation of gradient and area under the graph15.4Ex 15.4B8 Variable acceleration – part 1Start the lesson with a quiz on differentiation and integrations.Students use differentiation and integration to solve problems where acceleration is not constant. This lesson focuses on the use of differentiation.Use calculus to find displacement, velocity and acceleration when they are functions of timeQ4 – use calculus in kinematics for motion in a straight line: …Links back to Chapter 8 Differentiation and Chapter 9 Integration15.5Ex 15.5A9 Variable acceleration – part 2Students use differentiation and integration to solve problems where acceleration is not constant. This lesson focuses on the use of integration.Use calculus to find displacement, velocity and acceleration when they are functions of timeQ4 – use calculus in kinematics for motion in a straight line: …15.5Ex 15.5B10 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 15 in Student BookExam-style extension questions 15 in Student BookEnd-of-chapter test 15 on Collins ConnectCHAPTER 16 – Forces (8 Hours)Prior knowledge neededChapter 10Chapter 15TechnologyGraphics packages can be used to visualise problems.One hour lessonsLearning objectivesSubject content referencesTopic links Student Book references1 ForcesStudents are told the SI unit of force is the newton and mass is the kilogram. They use vectors, Pythagoras’ theorem and trigonometry to find resultant forces. They also use the face that if the resultant force is equal to zero then an object is in equilibrium.Find a resultant force by adding vectorsFind the resultant force in a given directionJ5 – use vectors to solve problems in pure mathematics and in context, including forcesP1 – understand and use derived quantities and units; force and weightR1 – understand the concept of a forceBuilds on Chapter 10 Vectors16.1EX 16.1A2 Newton’s laws of motion part oneStudents are told Newton’s laws of motion. They apply these to problems. Students are then introduced to the five different types of force. These are represented diagrammatically so that they are clearly distinguishable. Students then go on to use simplified models to represent real-life situation and solve problemsApply Newton’s laws and the equations of constant acceleration to problems involving forcesDistinguish between the forces of reaction, tension, thrust, resistance and weight Represent problems involving forces in a diagramR1 – understand the concept of a force; understand and use Newton’s first lawR2 – understand and use Newton’s second law for motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2-D vectors)R3 – understand and use weight and motion in a straight line under gravity16.2Ex 16.2A3 Newton’s laws of motion part twoStudents are told Newton’s laws of motion. They apply these to problems. Students are then introduced to the five different types of force. These are represented diagrammatically so that they are clearly distinguishable. Students then go on to use simplified models to represent real-life situation and solve problemsApply Newton’s laws and the equations of constant acceleration to problems involving forcesDistinguish between the forces of reaction, tension, thrust, resistance and weight Represent problems involving forces in a diagramR1 – understand the concept of a force; understand and use Newton’s first lawR2 – understand and use Newton’s second law for motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2-D vectors)R3 – understand and use weight and motion in a straight line under gravity16.2Ex 16.2B4 Vertical motionStudents move from particles travelling horizontally to particles travelling in vertically. When particles travel vertically they have to consider the effects of gravity and weight. Distinguish between the forces of reaction, tension, thrust, resistance and weightModel situations such as lifts, connected particles and pulleysR3 – understand and use weight and motion in a straight line under gravity; gravitational acceleration , g, and its value in S>I> units to varying degrees of accuracyR4 – understand and use Newton’s third law16.3Ex 16.3A5 Connected particles – part 1More complex problems are solved using Newton’s laws and simultaneous equations due to the consideration of resistive forces. Modelling assumptions are made and shown with diagrams when answering questions. The connected particles considered are particles suspended on ropes, vehicles towing caravans and lifts.Model situations such as lifts, connected particles and pulleysRepresent problems involving forces in a diagramR4 - understand and use Newton’s third law; equilibrium of forces on a particle and motion in a straight line; application to problems involving smooth pulleys and connected particles16.4Ex 16.4A6 Connected particles – part 2Recap on the chapter so far. Students complete exercise 16.4B.Model situations such as lifts, connected particles and pulleysRepresent problems involving forces in a diagramR4 - understand and use Newton’s third law; equilibrium of forces on a particle and motion in a straight line; application to problems involving smooth pulleys and connected particles16.4Ex 16.4B7 PulleysConnected particles are now considered in the form of pulleys. The main difference being that when an object is connected via a pulley it will be travelling in a different direction. Students should again draw diagrams representing the situation in the question. They solve problems using Newton’s laws of motion, the SUVAT equations and simultaneous equations.Model situations such as lifts, connected particles and pulleysRepresent problems involving forces in a diagramR4 - understand and use Newton’s third law; equilibrium of forces on a particle and motion in a straight line; application to problems involving smooth pulleys and connected particles16.5Ex 16.5A8 Chapter review lessonRecap key points of chapter. Students to complete the exam-style questions, extension questions, or online end-of-chapter testIdentify areas of strength and development within the content of the chapterSee belowAssessmentExam-style questions 16 in Student BookExam-style extension questions 16 in Student BookEnd-of-chapter test 16 on Collins Connect ................
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